Finite Group Databasebeta

← #29731013

Subgroups — #29731013

Order 15616.

916 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456278C211yesmaximal, characteristic, complemented
2C2×#16840024C212yesmaximal, complemented
3#21455283C211yesmaximal, characteristic
4C2×#16839967C211yesmaximal, characteristic, complemented
5#21454565C211yesmaximal, characteristic, complemented
6#21447121C212yesmaximal, complemented
7#21446935C212yesmaximal, complemented
8#21446663C211yesmaximal, characteristic, complemented
9#21446594C211yesmaximal, characteristic, complemented
10#21446335C212yesmaximal, complemented
11C4×#100×C61C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C2×#14519810C2212yes—
14C2×#14519803C2212yes—
15C22×#11758200C2212yes—
16C2×#14519785C2212yes—
17#16840312C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19#16840156—22no—
20#16840142—21no—
21#16840135—21no—
22#16840024—28no—
23#16839977—22no—
24#16839967—21no—
25#16839967—21no—
26#16839967—22no—
27#16839967—21no—
28#16839957—21no—
29#16839841—21no—
30#16839835—22no—
31C8×D244—21no—
32C2×#14519712C2212yes—
33#16839786C2211yescharacteristic
34C2×C4⋊#92418C2211yescharacteristic
35#16839761C2212yes—
36C2×#14519700C2212yes—
37C2×#14519695C2212yes—
38C2×#14519695C2212yes—
39#16839730C2212yes—
40#16839526C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#100×C61C2211yescharacteristic
43C2×#100×C61C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C4×M16×C61C2211yescharacteristic
46C61×#343C2211yescharacteristic
47C61×#343C2211yescharacteristic
48C2×#11758222C2312yes—
49C22×Dic122C2312yes—
50#14519810—24no—
51#14519810—24no—
52#14519810—42no—
53#14519810—42no—
54#14519803—24no—
55#14519803—42no—
56C2×Dic244—22no—
57C2×Dic244—22no—
58C2×#11758190—22no—
59C2×#11758190—22no—
60C2×#11758200—24no—
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200—42no—
65C2×#11758199—22no—
66#14519785—24no—
67#14519785—42no—
68C2×#11758194C2312yes—
69C2×#11758189—22no—
70C22×#92418C2312yes—
71C2×C8×D61—22no—
72#14519777—21no—
73#14519777—21no—
74#14519777—21no—
75#14519777—21no—
76#14519777D411yescharacteristic
77#14519774D411yescharacteristic
78#14519765D412yes—
79#14519751—21no—
80#14519751—21no—
81C4×D244—21no—
82C4×D244—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85#14519712—24no—
86C4⋊#92418—21no—
87C4⋊#92418—21no—
88C4⋊#92418D411yescharacteristic
89C4⋊#92418D412yes—
90C4⋊#92418D411yescharacteristic
91#14519700—22no—
92#14519700—22no—
93#14519700—22no—
94#14519697—22no—
95#14519695—24no—
96#14519695—42no—
97#14519689—22no—
98#14519689—22no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C2×M16×C61C2311yescharacteristic
102C2×M16×C61C2311yescharacteristic
103C2×M16×C61C2311yescharacteristic
104C2×M16×C61C2311yescharacteristic
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C42×C61C2311yescharacteristic
112#100×C61—22no—
113#100×C61D414yescomplemented
114#97×C61—21no—
115#97×C61—21no—
116C4×C8×C61—21no—
117C4×C8×C61—21no—
118C24×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—21no—
126C22×C4×C61—41no—
127C22×C4×C61—41no—
128C22×C4×C61—82no—
129C22×C4×C61#4111yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—22no—
133C22×Dic61—22no—
134#11758222—22no—
135#11758222—22no—
136#11758222—22no—
137#11758222—22no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244C2×D412yes—
141C2×C4×D61—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122C2×D412yes—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152M16×C61—21no—
153M16×C61—21no—
154M16×C61C2×D412yes—
155M16×C61C2×D412yes—
156M16×C61#4112yes—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C22⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169#11758200—44no—
170#11758200—44no—
171#11758200—44no—
172#11758199—44no—
173#11758198—22no—
174#11758198—22no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—41no—
178C4⋊Dic61—41no—
179C4⋊Dic61—41no—
180C4⋊Dic61—41no—
181#11758194—22no—
182#11758194—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418C2×D412yes—
187C2×#92418C2×D412yes—
188Dic244—44no—
189Dic244—44no—
190#11758190—44no—
191#11758190—44no—
192#11758189—44no—
193C8×D61—44no—
194C23×C61—22no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—81no—
198C23×C61—81no—
199C23×C61C2×#4111yescharacteristic
200C22×D61—22no—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61—41no—
204D4×C61—41no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—82no—
209C2×C4×C61—82no—
210C2×C4×C61—81no—
211C2×C4×C61#12111yescharacteristic
212C2×C4×C61#12111yescharacteristic
213C2×C4×C61#12111yescharacteristic
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)11yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C22⋊D61—42no—
222C22⋊D61—42no—
223C2×Dic61—22no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227D244—22no—
228D244—22no—
229D244—42no—
230D244—81no—
231C4×D61—42no—
232C4×D61—42no—
233Dic122—22no—
234Dic122—22no—
235Dic122—42no—
236Dic122—42no—
237Dic122—42no—
238Dic122—42no—
239C8×C61—22no—
240C8×C61—22no—
241C8×C61—22no—
242C8×C61—41no—
243#92418—22no—
244#92418—22no—
245#92418—42no—
246#25405—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—42no—
250C22×C61—82no—
251C22×C61—82no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61C2×#12111yescharacteristic
255C22×C61#53211yescharacteristic
256C22×C61#54011yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—81no—
264C4×C61#47112yescomplemented
265C4×C61D4211yescharacteristic
266C4×C61D4211yescharacteristic
267C4×C61#54611yescharacteristic
268C4×C61#55011yescharacteristic
269Dic61—42no—
270Dic61—82no—
271#3152—611no—
272C2×#471—612no—
273#3036—611no—
274C2×#461—611no—
275#2935—611no—
276#2052—612no—
277#2019—612no—
278#1966—611no—
279#1950—611no—
280#1908—612no—
281C4×#100D6111yescharacteristic, complemented
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—82no—
285C2×C61#242811yescharacteristic
286C2×C61C2×#47111yescharacteristic
287C2×C61#306911yescharacteristic
288D61—82no—
289C2×#137—612no—
290C2×#137—612no—
291C2×AGL(1,ℤ/8)—612no—
292C22×SD16—612no—
293C2×#131—612no—
294#532—611no—
295C2×C4⋊Q8—611no—
296C2×C4×D4C2×D6111yescharacteristic
297#497—1222no—
298#494—1221no—
299#492—1221no—
300#471—1228no—
301#463—1222no—
302#461—1221no—
303#461—1222no—
304#461—1221no—
305#461—1221no—
306#459—1221no—
307#436—1221no—
308#435—1222no—
309C8×D4—1221no—
310C2×#108—612no—
311#423—611no—
312C2×C4⋊C8—611no—
313#417—612no—
314C2×#103—612no—
315C2×#101—612no—
316C2×#101—612no—
317C2×#100C2×D6111yescharacteristic
318C2×#100C2×D6111yescharacteristic
319#408—612no—
320C4×M16C2×D6111yescharacteristic
321C4×M16C2×D6111yescharacteristic
322#343C2×D6111yescharacteristic
323#343C2×D6111yescharacteristic
324#341—612no—
325C61#2540511yescharacteristic, complemented, Sylow
326C2×#41—612no—
327C22×Q8—612no—
328C22×D4C22×D6111yescharacteristic
329C23×C4—21no—
330#137—1224no—
331#137—1224no—
332#137—2442no—
333#137—2442no—
334AGL(1,ℤ/8)—1224no—
335AGL(1,ℤ/8)—2442no—
336C2×Q16—1222no—
337C2×Q16—1222no—
338C2×SD16—1222no—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1222no—
343C2×SD16—1224no—
344C2×SD16—2442no—
345C2×D8—1222no—
346#131—1224no—
347#131—2442no—
348C2×M16—612no—
349C2×M16—1222no—
350C2×M16C22×D6111yescharacteristic
351C2×M16C22×D6111yescharacteristic
352C2×M16C22×D6111yescharacteristic
353C2×M16C22×D6111yescharacteristic
354C22×C8—612no—
355C22×C8—1222no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361#127—611no—
362#124—612no—
363#121—1221no—
364#121—1221no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—1221no—
370C4×D4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C42C22×D6111yescharacteristic
376#108—1224no—
377C4⋊C8—611no—
378C4⋊C8—612no—
379C4⋊C8—611no—
380C4⋊C8—1221no—
381C4⋊C8—1221no—
382#103—1222no—
383#103—1222no—
384#103—1222no—
385#102—1222no—
386#101—1224no—
387#101—2442no—
388#100—22no—
389#100D24414yescomplemented
390C22⋊C8—1222no—
391C22⋊C8—1222no—
392#97—21no—
393#97—21no—
394C4×C8—21no—
395C4×C8—21no—
396C24—21no—
397#41—1222no—
398#41—1222no—
399#41—1222no—
400#41—1222no—
401C2×Q8—612no—
402C2×Q8—1222no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—612no—
414C2×D4—1222no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4C2×D24412yes—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—82no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4#1175822211yescharacteristic
425C22×C4#1175822211yescharacteristic
426C22×C4C23×D6111yescharacteristic
427Q16—2444no—
428Q16—2444no—
429SD16—2444no—
430SD16—2444no—
431SD16—2444no—
432SD16—2444no—
433SD16—2444no—
434D8—2444no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—21no—
439M16—21no—
440M16—1222no—
441M16—1222no—
442M16—2444no—
443M16C2×D24412yes—
444M16C2×D24412yes—
445M16#1175822212yes—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—612no—
451C2×C8—612no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—2444no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—1222no—
467C22⋊C4—1222no—
468C42D4×D6111yescharacteristic
469C42D4×D6111yescharacteristic
470C42D4×D6111yescharacteristic
471C42D4×D6111yescharacteristic
472C23—22no—
473C23—41no—
474C23—41no—
475C23—81no—
476C23—81no—
477C23—1222no—
478C23C2×#1175822211yescharacteristic
479Q8—1222no—
480Q8—1222no—
481Q8—2442no—
482Q8—2442no—
483Q8—2442no—
484Q8—2442no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—1222no—
490D4—1222no—
491D4—4881no—
492D4—2442no—
493D4—2442no—
494D4—2442no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—1222no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4#1451975011yescharacteristic
508C2×C4#1451975011yescharacteristic
509C2×C4#1451975011yescharacteristic
510C2×C4#1451975011yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C22×D24411yescharacteristic
514C2×C4C2×#1175822211yescharacteristic
515C2×C4#1451982311yescharacteristic
516C2×C4#1451982311yescharacteristic
517C8—22no—
518C8—22no—
519C8—22no—
520C8—41no—
521C8—1222no—
522C8—1222no—
523C8—2442no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—82no—
528C22—82no—
529C22—81no—
530C22—81no—
531C22—2442no—
532C22—2442no—
533C22—2442no—
534C22C2×#1451975011yescharacteristic
535C22#1684031211yescharacteristic
536C22#1684036711yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—81no—
541C4—4882no—
542C4—2442no—
543C4#1684002412yescomplemented
544C4D4×D24411yescharacteristic
545C4D4×D24411yescharacteristic
546C4#1684040811yescharacteristic
547C4#1684043311yescharacteristic
548C2—21no—
549C2—21no—
550C2—82no—
551C2—4882no—
552C2#2144940711yescharacteristic
553C2C2×#1684002411yescharacteristic
554C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#100×C61C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenterC4×#100D61
hypocenterC61#25405
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21456278C2semidirect
2. C2×#16840024C2semidirect
4. C2×#16839967C2semidirect
5. #21454565C2semidirect
6. #21447121C2semidirect
7. #21446935C2semidirect
8. #21446663C2semidirect
9. #21446594C2semidirect
10. #21446335C2semidirect
11. C4×#100×C61C2semidirect
113. #100×C61D4semidirect
264. C4×C61#471semidirect
281. C4×#100D61semidirect
325. C61#25405semidirect
389. #100D244semidirect
543. C4#16840024semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214562782
C2×#168400242
#214552832
C2×#168399672
#214545652
#214471212
#214469352
#214466632
#214465942
#214463352
C4×#100×C612
#2540561